Search: id:A082758
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%I A082758
%S A082758 1,3,19,141,1107,8953,73789,616227,5196627,44152809,377379369,
%T A082758 3241135527,27948336381,241813226151,2098240353907,18252025766941,
%U A082758 159114492071763,1389754816243449,12159131877715993,106542797484006471
%N A082758 Sum of the squares of the trinomial coefficients (A027907).
%C A082758 a(n) = T(2n,2n) = coefficient of x^(2n) in (1+x+x^2)^(2n), T is the trinomial 
               triangle A027907; Integral representation : a(n) = 1/Pi Integral[(1+2x)^(2n)/
               Sqrt[1-x^2],{x,-1, 1}], i.e. a(n) is the moment of order 2n of the 
               random variable 1+2X, where the distribution of X is an arcsin law 
               on the interval (-1,1). - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), 
               Jan 22 2008
%F A082758 a(n) = Sum[ T(n, k)^2, {k, 0, n} ] where T(n, k) = trinomial coefficients 
               (A027907).
%F A082758 a(n)=sum(k=0, n, binomial(2*n-k, k)*binomial(2*n, k)) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Jul 30 2003
%F A082758 G.f.: (1/sqrt(1+2x-3x^2)+1/sqrt(1-2x-3x^2))/2 (with interpolated zeros) 
               - Paul Barry (pbarry(AT)wit.ie), Jan 04 2005
%F A082758 a(n)=sum{k=0..n, C(2n,2k)*C(2k,k)}=sum{k=0..n, C(n+k,2k)*C(2n,n+k)}; 
               [From Paul Barry (pbarry(AT)wit.ie), Dec 16 2008]
%o A082758 (PARI) a(n)=sum(k=0,n,binomial(2*n-k,k)*binomial(2*n,k))
%Y A082758 Bisection of A002426.
%Y A082758 Sequence in context: A074559 A027314 A025571 this_sequence A110525 A058859 
               A095002
%Y A082758 Adjacent sequences: A082755 A082756 A082757 this_sequence A082759 A082760 
               A082761
%K A082758 easy,nonn
%O A082758 0,2
%A A082758 Emanuele Munarini (munarini(AT)mate.polimi.it), May 21 2003

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